Schwinger-boson approach to the kagome antiferromagnet ... - CNRS

PHYSICAL REVIEW B 81, 064428 共2010兲

Schwinger-boson approach to the kagome antiferromagnet with Dzyaloshinskii-Moriya interactions: Phase diagram and dynamical structure factors L. Messio,1 O. Cépas,2 and C. Lhuillier1 1

Laboratoire de physique théorique de la matière condensée, UMR 7600 CNRS, Université Pierre-et-Marie-Curie-Paris 6, 75252 Paris Cedex 05, France 2Institut Néel, CNRS et Université Joseph Fourier, BP 166, F-38042 Grenoble Cedex 9, France 共Received 14 December 2009; revised manuscript received 4 February 2010; published 24 February 2010兲 We have obtained the zero-temperature phase diagram of the kagomé antiferromagnet with DzyaloshinskiiMoriya interactions in Schwinger-boson mean-field theory. We find quantum phase transitions 共first or second order兲 between different topological spin-liquid and Néel-ordered phases 共either the 冑3 ⫻ 冑3 state or the so-called Q = 0 state兲. In the regime of small Schwinger-boson density, the results bear some resemblances with exact diagonalization results and we briefly discuss some issues of the mean-field treatment. We calculate the equal-time structure factor 共and its angular average to allow for a direct comparison with experiments on powder samples兲, which extends earlier work on the classical kagomé to the quantum regime. We also discuss the dynamical structure factors of the topological spin-liquid and the Néel-ordered phases. DOI: 10.1103/PhysRevB.81.064428

PACS number共s兲: 75.10.Jm, 75.40.Mg, 75.50.Ee

I. INTRODUCTION

The Dzyaloshinskii-Moriya 共DM兲 interactions1 are inevitably present in S = 1 / 2 magnetic oxides when the magnetic bonds lack inversion centers, which is the case of the kagome lattice. Although small in strength 共it originates in the spin-orbit coupling兲, such a correction may favor other phases than the ones usually predicted by using the standard Heisenberg model. By breaking explicitly the spin-rotation symmetry of the system, the Dzyaloshinskii-Moriya forces tend to reduce the spin fluctuations and may therefore be detrimental to the long-searched spin-liquid phases. However, would the Heisenberg phase be gapped, such as in valence bond crystals 共generalized spin-Peierls states兲 or in topological spin liquids 共TSLs兲,2–4 then it would be robust against perturbations typically smaller than the gap. An example is given by the Shastry-Sutherland compound SrCu2共BO3兲2 which remains in the singlet phase in the presence of Dzyaloshinskii-Moriya couplings.5 In the kagome antiferromagnet 共with pure nearest-neighbor Heisenberg interaction兲, the very existence of a gap remains an open question,6 algebraic spin-liquid and gapless vortex phases have been proposed as alternatives in the recent years.7–9 Current numerical estimates of the gap provide a small upper bound ⬃0.05J.6,10–13 In any case the gap 共if it exists兲 could be smaller than the Dzyaloshinskii-Moriya coupling 共especially in copper oxides where it is typically ⬃0.1J兲 and the latter is therefore particularly relevant. Experiments on the kagome compound ZnCu3共OH兲6Cl3 共Refs. 14–17兲 have found no spin gap 共despite a temperature much lower than the upper estimation of the gap兲 but the chemical disorder18–20 and, precisely, the existence of DzyaloshinskiiMoriya interactions21–24 make the spin gap issue not yet clear. In fact, the Dzyaloshinskii-Moriya interactions were argued to induce long range Q = 0, 120° Néel order in the kagome antiferromagnet: the algebraic spin-liquid theory predicts the instability at a critical strength Dc = 0 共Ref. 25兲 while there is a finite quantum critical point at Dc ⬃ 0.1J in 1098-0121/2010/81共6兲/064428共8兲

exact diagonalization results on samples of size up to N = 36.23 Since it is clear that there is no Néel order at D strictly zero10,26 and there is Néel order for D large enough,23 it is tempting to tackle the problem using the Schwingerboson representation for the spin operators.27 Indeed this approach allows in principle to capture both topological spinliquid and Néel-ordered phases28 and offers a first framework to describe this quantum phase transition. The caveat is that the actual Schwinger-boson mean-field solution for the S = 1 / 2, D = 0 kagome antiferromagnet is already long-ranged ordered 共LRO兲 and it is only at smaller values of S 共which in this approach is a continuous parameter兲 that a disordered spin-liquid phase is stabilized. This result may however be an artifact of the mean-field approach and it is possible that fluctuations not taken into account at this level do stabilize the disordered phase for the physical spin-1/2 system. It is therefore interesting to see what phases the Schwinger-boson mean-field theory predicts for the kagome antiferromagnet perturbed by Dzyaloshinskii-Moriya interactions. In Sec. II we present the model and the method. In Sec. III we discuss the phase diagram together with general ground-state properties. In Sec. IV, we illustrate the evolution of observables across the quantum phase transition from topological spin liquid to long-ranged Néel order: the spinon spectrum, the gap, the order parameter, and the condensed fraction of bosons. We calculate the equal-time structure factor, its powder average, and briefly compare both to classical calculation and experimental results. In Sec. V we present the dynamical spin structure factor and its behavior in the two phases. We also describe how these behaviors emerge from the properties of the spinon spectrum. II. MODEL

We have considered additional DM interactions to the standard Heisenberg model on the kagome lattice H = 兺关JSi · S j + Dij · 共Si ⫻ S j兲兴, 具i,j典

共1兲

where 具i , j典 stands for nearest neighbors 共each bond is counted once兲 and Si is a quantum spin operator on site i, the

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©2010 The American Physical Society


MESSIO, CÉPAS, AND LHUILLIER

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FIG. 1. The Dzyaloshinskii-Moriya field in the model 共spin rotated frame—see text兲. 1

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1 A†ij ⬅ 共ei␪ija†i b†j − e−i␪ijb†i a†j 兲 2

共3兲

with ␪ij = Dij / 共2J兲. A similar approach has been developed by Manuel et al.29 on the square lattice. With this definition and up to small corrections of order D2ij / J, the model takes its standard form27 H = − 2J 兺 A†ijAij + NzJS2/2, 具i,j典

共4兲

where N is the number of lattice sites and z = 4 the coordination number. H has a local U共1兲 gauge symmetry: it is invariant under 共ai , bi兲 → 共ei␣iai , ei␣ibi兲, where ␣i 苸关0 , 2␲兴. Applied to the vacuum of boson, A†ij creates a superposition of a singlet and a triplet state on the ij bond, i.e., the exact ground state of a single bond of Eq. 共4兲. In mean-field theory,27,30 the quartic Hamiltonian 共4兲 is replaced by a self-consistent quadratic Hamiltonian with a bond varying order parameter Aij ⬅ 具Aij典 and the constraint ni = 2S enforced on average with Lagrange multipliers ␭i. Up to a constant the mean-field Hamiltonian reads

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FIG. 2. 共Color online兲 The bond order parameter Aij of the four symmetric Ansätze of model Eq. 共1兲. Its modulus A is a constant for all bonds, an arrow from i to j means Aij ⬎ 0. The fluxes through the hexagon and the rhombus 共⌽Hex , ⌽Rho兲 are in 共a兲共0,0兲, 共b兲共␲ , 0兲, 共c兲共0 , ␲兲, and 共d兲共␲ , ␲兲. The unit cell is shown by dashed lines 共twice larger for the last two Ansätze兲.

HMF = − 2J 兺 AⴱijAij + A†ijAij − 兩Aij兩2 − 兺 ␭i共ni − 2S兲. 具i,j典

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where a†i 共respectively, b†i 兲 creates a boson on site i with spin ↑ 共respectively, ↓兲 and ␴ are the Pauli matrices. To fix the length of the spin, S2i = 共ni / 2兲共ni / 2 + 1兲 = S共S + 1兲, we need to have ni = 2S bosons per site. We define the bond creation operator

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Dzyaloshinskii-Moriya field Dij is taken to be along the z axis and is staggered from up to down triangles 共see Fig. 1兲. For this, we work in a rotated frame which allows to eliminate the other components 共the Si are to be viewed as rotated operators兲.23 In this case, the Dzyaloshinskii-Moriya field favors a vector chirality along z and the model 共1兲 has a global U共1兲 symmetry which can be spontaneously broken or not. The Schwinger-boson representation is written

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共5兲 We now restrict our search to solutions whose physical 共thus gauge invariant兲 observables do not break the space-group symmetry of the Hamiltonian and hence could realize spinliquid states. There are only four classes of such states 共called Ansätze in Refs. 28 and 31 and in the following兲, labeled by their projective symmetry group,31 or equivalently by fluxes around hexagons and rhombus 共⌽Hex , ⌽Rho兲 = 共0 , 0兲, 共␲ , 0兲, 共0 , ␲兲, or 共␲ , ␲兲. The flux ⌽ around a loop 共i1 , i2 , . . . , i2n兲 with an even number of links is defined by32 ⴱ ⴱ Kei⌽ = A12共− A23 兲, . . . ,A2n−1,2n共− A2n,1 兲.

共6兲

It is a gauge-invariant quantity. In each of the four Ansätze, all Aij have the same amplitude 兩Aij兩 = A and are real in a well chosen gauge. Their signs are represented on Fig. 2. These four Ansätze are identical to those obtained by Wang and Vishwanath for the kagome Heisenberg model: they are fully determined by rotational and translational invariances of the spin Hamiltonian on the lattice. Note that to be invariant under the lattice symmetries, the bosonic Hamiltonian of these Ansätze must be gauge transformed, which does not modify the corresponding spin Hamiltonian. That explains the noninvariance of these Ansätze under the lattice symmetries. The 共0,0兲 Ansatz corresponds to the 冑3 ⫻ 冑3 state and the 共␲ , 0兲 Ansatz to the Q = 0 state, originally found by Sachdev28 while the last two involve larger unit cells and may as well be relevant for longer range interaction or ring exchange.31,33 Using the translation symmetry of the mean-field Ansätze, the Hamiltonian is Fourier transformed,

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SCHWINGER-BOSON APPROACH TO THE KAGOME…

H MF = 兺 ␾q† Nq␾q + NJzA2 + 共2S + 1兲N␭

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Pq =

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Ansatz (0,0) Ansatz (π,0) Ansatz (0,π) Ansatz (π,π)

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q␮

where ␻q␮ is the dispersion relation of the ␮ = 1 , . . . , 2m spinon modes. Each mode is twice degenerate because of the ˜ 典 is the vacuum time-reversal symmetry. The ground state 兩0 of the Bogoliubov bosons. At zero temperature, A and ␭ are determined by extremizing the total energy, subject to the constraints, A = 兩具Aij典兩

具ni典 = 2S

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关the energy is in fact a saddle point, minimum in A and maximum in ␭ 共Ref. 35兲兴. III. PHASE DIAGRAM

To obtain the phase diagram, the two self-consistent Eq. 共12兲 are implemented numerically for each of the four Ansätze of Fig. 2. The energies of the different Ansätze are shown in Fig. 3 versus ␪ = D / 共2J兲 for three values of S: 0.025, 0.2, and 0.5. The corresponding full phase diagram of the model is shown in Fig. 4. Before discussing the predictions of this model, let us remark that in the small S limit all these phases can in fact be captured by an analytic perturbative expansion in term of flux through closed loops.32 At small S, the density of bosons is small and the constraints 关Eq. 共12兲兴 imply that A must be small compared to ␭. The mean-field energy can then be expanded in terms of gauge-invariant products of bond order parameters Aij along closed loops. Following Tchernyshyov et al.,32 we have computed the expansion up to loops of length 16 in order to calculate the energy difference between the four Ansätze. The results of these calculations give the low S phase boundaries 共dashed lines superimposed to the exact results in Fig. 4兲. It is seen on this example that the so-called flux expulsion conjecture32 which predicts that the ground state in nonfrustrated models has zero flux through

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FIG. 3. 共Color online兲 Ground state energies per site of the four Ansätze vs ␪ = D / 2J, for S = 0.025, 0.2, and 0.5.

any closed loop does not apply to frustrated problem, where the 共0 , ␲兲 and 共␲ , 0兲 appear as ground states in an extended range of parameters. For the sake of clarity, we will first discuss the full phase diagram 共Fig. 4兲. 共We postpone to the next paragraph the illustration on the spinon spectrum of the differences between topological spin-liquid and Néel-ordered phases. Let us only mention at this stage that a spin-liquid phase is characterized by a fully gapped spinon spectrum, whereas in a Néel-ordered phase the spinon spectrum is gapless at the thermodynamic limit, as the magnon modes.兲 For S = 1 / 2, there is a direct first-order transition between the long-ranged Néel-ordered 冑3 ⫻ 冑3 and Q = 0 phases for a finite . . . . .

.

.

.

.

.

FIG. 4. Phase diagram at zero temperature 关spin S , ␪ = D / 共2J兲兴: TSL and Néel LRO phases characterized by their fluxes through hexagons and rhombi. For larger S the region of existence of the 共0,0兲 phase shrinks. Dashed lines are the result of a perturbative expansion at small S 共see text兲.

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Dzyaloshinskii-Moriya coupling. This finite range of existence of the 冑3 ⫻ 冑3 phase shrinks with increasing values of the spin, which is fully compatible with the classical solution.36 For low S values, and increasing ␪, Figs. 3 and 4 show a sequence of first-order transitions between the 共0,0兲共冑3 ⫻ 冑3 short-range fluctuations兲, 共0 , ␲兲, and 共␲ , 0兲共Q = 0, short-range fluctuations兲 spin-liquid phases. 关The 共␲ , ␲兲 state is always at higher energy and never realized.兴 The 共0 , ␲兲 state was argued to be stabilized by four-spin interactions up to a large critical S.31 It also appears here in a small part of the phase diagram for very small S but first-order transitions prevent its stability at larger S. In the absence of Dzyaloshinskii-Moriya coupling, the S = 1 / 2 results of this approach are qualitatively not consistent with exact diagonalizations which point to a nonmagnetic phase. But in the range of parameters around S ⬃ 0.2, the 共␲ , 0兲 Schwinger-boson mean-field results are qualitatively not very far from exact diagonalization results: there are short-ranged Q = 0 correlations in the Heisenberg case12,37 and a second-order phase transition to 120° Q = 0 Néel order under the effect of Dzyaloshinskii-Moriya coupling.23 As already mentioned, it may be that a theory beyond mean field, with a better treatment of the constraint shifts the region S ⬃ 0.2 to the physically accessible S = 1 / 2. Indeed in the Schwinger bosons mean-field approach it is well known27 that there are large fluctuations of the number of bosons. As a consequence, the square of the spin operator 具S2i 典 = S共S + 1兲 + 共具n2i 典 − 具ni典2兲/4

共13兲

takes a value 3/2 times larger than S共S + 1兲共at D = 0兲. The prefactor is even larger in the presence of DzyaloshinskiiMoriya interaction and amounts to ⬃1.75 for ␪ = 0.25. From the physical spin-1/2 point of view, the mean-field approximation leads to 共unwanted兲 extra fluctuations and, on average, the spin length is larger than assumed 关because of 共13兲兴. The SU共2兲 symmetry of the Heisenberg model can be generalized to Sp共N兲关which reduces to SU共2兲 when N = 1兴 by duplicating N times each pair 共ai , bi兲 of boson operators: 共ai␣ , bi␣兲, where ␣ = 1 ¯ N is a “flavor” index.2 It can be shown that the different boson flavors decouple in the limit of large N, leading to N uncorrelated copies of the single flavor problem, for which the exact solution is given by the present mean-field treatment of the N = 1 model. Thus, it is only in the N = ⬁ limit of the model that the mean-field treatment becomes exact and that the fixed “spin length” is recovered. However, the mean-field theory can describe qualitatively the magnetically ordered phases and the deconfined Z2 spin-liquid phases of the SU共2兲 model.39

(b)

(a)

FIG. 5. Spinon dispersions along ⌫-K-M-⌫ of the first Brillouin Zone 共see Fig. 6 for the definition兲 for the 共␲ , 0兲 Ansatz, S = 0.2. Left: ␪ = 0, the system has a gap and is a topological spin liquid with short-range Q = 0 correlations. The lower band is twofold degenerate corresponding to two different chiralities. Right: ␪ = 0.25, the lower branch becomes gapless at ⌫ in the thermodynamic limit and gives rise to the Goldstone mode of the long-range Néel order.

of such a spectrum for the Ansatz 共␲ , 0兲. The spinon band structure is shown in the first Brillouin zone, it has a gap of order O共1兲 at point ⌫ indicative of short-range Q = 0 correlations. This gap does not go to zero in the thermodynamic limit: this phase is a disordered topological spin liquid. Adding a Dzyaloshinskii-Moriya perturbation has two effects on the spectrum. It lifts the degeneracy of the lower band corresponding to the symmetry breaking of the model from SU共2兲 to U共1兲 and it reduces the gap of the spinon mode that has the chirality opposite to that of the DzyaloshinskiiMoriya field.

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B. Bose-Einstein condensation

With increasing S or ␪, the gap decreases and above a critical spin Sc共␪兲, the spinon spectrum shows a finite-size gap, which collapses with system size N as O共1 / N兲. Such a spectrum is shown in Fig. 5 共right兲. In the thermodynamic ˜q l ␴ limit, the bosons condense in the soft mode 共noted ␾ 00 41 with ␴ = ↑ , ↓兲. This gives a macroscopic contribution to the total number of Schwinger bosons, 兩Vqij兩2 兩Vqij兩2 1 = x q0N + 兺 . 共14兲具ni典 = N = 兺兺 S S 2S i qij q⫽q0,ij The condensed fraction xq0 in the soft mode is of order O共1兲共or equivalently 兩Vq0il0兩 ⬃ 冑N兲. The transition to this BoseEinstein condensed phase corresponds to the development of long-range antiferromagnetic correlations, as can be seen by computing the static structure factor, Sxx共Q兲 =

IV. SPINON SPECTRUM AND QUANTUM PHASE TRANSITION FROM A TOPOLOGICAL SPIN-LIQUID TO A NÉEL-ORDERED PHASE A. Spin-liquid phases (low S)

In the low S regime, the spinon spectrum of Eq. 共11兲 is gapped everywhere. Figure 5 共left兲 gives a typical example

3 兺 eiQ·共Ri−R j兲具0˜ 兩Sxi Sxj 兩0˜ 典, 4N i,j

共15兲

where Ri is the position of site i and x is an axis in the easy plane perpendicular to Dij. The difference in static structure factor between topological spin-liquid and Néel-ordered phases is illustrated in Fig. 6 for the 共␲ , 0兲 Ansatz across the Bose-Einstein condensation. In the spin-liquid phase, the structure factor has broad

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S(|Q|)

1.5

(a)

0.5

(b)

FIG. 6. 共Color online兲 Static structure factor Sxx in the extended Brillouin zone for the 共␲ , 0兲 state 共Q = 0兲 for a lattice size N = 1296. In the spin-liquid phase TSL␲,0 共left, S = 0.2, ␪ = 0兲 there are broad features about Me which become peaks with divergent intensity in the ordered phase LRO␲,0 共right, S = 0.2, ␪ = 0.25兲.

features located at Q = Me 共Fig. 6, left兲 and at equivalent reciprocal points 共these are the ⌫ points of the second Brillouin zone兲. This structure factor looks very similar to exact diagonalization and density matrix renormalization group 共DMRG兲 results.12,37 The features become sharp in the BoseEinstein condensed phase 共Fig. 6, right兲, where Sxx共Me兲 becomes proportional to N, 3 xx Cst 2 + S 共Me兲 = mAF 冑N + ¯ , 4N

共16兲

2 is the order parameter corresponding to longwhere mAF range correlations of the 120° Q = 0 Néel type.42 We have 2 by fitting the numerical results 共up to N extracted mAF = 1764兲 to Eq. 共16兲 with finite-size corrections up to order 1 / N.43 The extrapolation to the thermodynamic limit of mAF, together with the condensate fraction xq0 and the gap of the soft spinon are given in Fig. 7 for S = 0.2. While near the second-order phase transition, the extrapolation of the condensed fraction behaves very smoothly and vanishes right at the point where the gap opens, the extrapolation of the order parameter gives a small shift. We note that mAF is very small 共a few percent兲 in this range of parameters and, therefore, more accurate extrapolations would require using larger system sizes close to the critical point. For strong enough Néel order, however, the two order parameters are clearly proportional 共mAF ⬀ xq0兲.

0.05

0 0

0.05

0.1

θ

0.15

0.2

10 5

0 0

0

1

0.25

FIG. 7. 共Color online兲 Gap of the soft mode, order parameter mAF, and condensed fraction, xq0 extrapolated to the thermodynamic limit for the 共␲ , 0兲 Ansatz as a function of ␪ = D / 共2J兲共S = 0.2兲.

2

|Q| (in units of |Me|)

3

FIG. 8. 共Color online兲 Static structure factor 共powder averaged兲 across the quantum phase transition from the spin liquid to the Néel phase 关S = 0.2 and ␪ = 0 共solid, black兲, 0.05 共dash, blue兲, and 0.2 共dot, red兲兴. Inset: S = 1 / 2, ␪ = 0. Small oscillations are finite-size effects.

We have also calculated the static structure factor for powder samples 关denoted by S共兩Q兩兲兴 by averaging Eq. 共15兲 over all directions of Q. In the 共␲ , 0兲 spin-liquid phase 共Fig. 8 at small D兲, the overall shape is characteristic of shortrange correlations of liquids. We can compare with the calculation of the diffuse scattering for the classical spin-liquid kagomé antiferromagnet by Monte Carlo simulation.44 Here the position of the first broad peak is at 兩Me兩 instead of 兩Ke兩共and the second broad feature is at 冑7兩Me兩兲. This simply reflects the difference of short-range correlations of the 共␲ , 0兲 Ansatz and the 冑3 ⫻ 冑3 classical spin liquid. In addition, compared with classical Monte Carlo simulations, we find no intermediate shoulder between the two main broad peaks 共except for a little hump at 冑3兩Me兩兲, a point which seems in fact to be closer to recent experiments on a spinliquid deuteronium jarosite.45 Moreover, since the response is due to quantum fluctuations we expect a rather weak sensitivity to the temperature up to temperatures of the order of a fraction of J. When D increases, we see the development of the Bragg peaks in the ordered phase, which increase as the square of the order parameter when we go deep into the ordered phase 关Fig. 8 共inset兲兴. In the ordered phase we can identify two distinct contributions to Eq. 共15兲 by using the sum rule Sxx共Q兲 =

Gap x0 mAF

0.1

1

1 2␲

冕

d␻Sxx共Q, ␻兲.

共17兲

There are the Bragg peaks 共␻ = 0兲 and also the inelastic collective modes 共which we will detail below兲 which give the additional magnetic background scattering 共which is the only contribution to scattering in the spin-liquid phase兲. It is noteworthy that the latter is relatively strong for low spin 共Fig. 8兲 and becomes relatively much smaller once the order parameter is large 共inset of Fig. 8兲. In fact the transfer of spectral weight from the magnetic continuum background to the Bragg peaks goes as the square of the order parameter. Note also that S共兩Q兩兲 does not vanish any more for small Q at D ⫽ 0, this is because in the presence of the anisotropy the ground state is no longer an SU共2兲 singlet. Although the effect is small the measurement at small 兩Q兩 in the spin-

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FIG. 9. 共Color online兲 Dynamical structure factors Sxx共q , ␻兲 and Szz共q , ␻兲 for the Q = 0 Ansatz in the spin-liquid phase, S = 0.1 or with long-range order, S = 0.2. The system size is N = 576. ␪ is taken to be 0.25 to emphasize the anisotropy of Sxx共q , ␻兲 and Szz共q , ␻兲. In the Néel phase, the largest peak 共in blue online兲 is the quasielastic peak, next in intensity are the magnon branches 共in red online兲共intensities are cut to see the weaker continuum兲.

liquid phase could help to figure out what the anisotropy is 共or give an upper bound when the signal is small, see, e.g., Ref. 46兲. In the ordered phase, the finite uniform susceptibility should give a finite contribution at Q = 0 but we recall that this contribution is absent for the U共1兲 singlet ground state of Schwinger-boson theory.

V. DYNAMICAL SPIN STRUCTURE FACTOR

The Schwinger-boson approach allows to calculate the dynamical response of the system, which is interesting both theoretically and for a direct comparison with experiments. The inelastic neutron cross section is proportional to the spin dynamical structure factor S␣␣共Q, ␻兲 =

冕

+⬁

␣ ␣ dtei␻t具S−Q 共t兲SQ 共0兲典,

共18兲

−⬁

␣ ˜ 2 兩0典兩 ␦共␻ − ␻ p兲, =2␲ 兺兩具p兩SQ

共19兲

p

where ␣ = x , y , z depending on the polarization of the incident ˜ 典 is the ground state, and neutrons, 兩0

SQ =

冑

3 兺 e−iQ·RixSix 4N xi

共20兲

with Rix the position of the site ix. We use the Fourier transform and the Bogoliubov transformation to express SQ in terms of quasiparticle operators 关Eq. 共10兲兴. At zero tempera˜ 典 is the vacuum of quasiparticles, only creation ture, since 兩0 operators are retained. Given that Six is quadratic in boson operators, we can only create spinons by pair. For example, the following term is present in SzQ: ⴱ † ˜ ˜ lq兲†共b Uqil V共q+Q兲in共a 兺 n−共q+Q兲兲 , qln

共21兲

˜ 典 in the matrix element 关Eq. 共19兲兴 creates which applied to 兩0 a pair of spinons 兩p典 with energy ␻ p = ␻ql + ␻−共q+Q兲n and wave vector −Q. The intensity of the transition is obtained by computing the product of matrices such as in Eq. 共21兲 for each pair of modes. We now discuss the general features of the dynamical spin structure factor and show the results for the Q = 0 Ansatz in Fig. 9. In the spin-liquid phase 关S ⬍ Sc共␪兲兴, all spinons are gapped and the two-spinon excitations form a gapped continuum, the bottom edge of which is given by the minimum of ␻ql + ␻−共q+Q兲n over all q and all spinon bands 共l , n兲. Sxx共Q , ␻兲 and Szz共Q , ␻兲 are given in Figs. 9共a兲 and 9共b兲. In

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these figures, we have taken D large enough to see the anisotropy of the response 共and a small S = 0.1 to be in the spin-liquid phase兲. When S increases the lower edge of the continuum shifts to zero and its intensity increases continuously. For S ⬎ Sc共␪兲关Figs. 9共c兲 and 9共d兲, S = 0.2, ␪ = 0.25兴, due to ˜ q l ␴ 关see Fig. 5 共right兲兴 the system the soft spinon mode ␾ 00 enters a Bose-condensed phase: it has long-range correlations and low-energy excitations varying as 1 / N, at Me. This spinon mode has a singular contribution ⬃冑N 关see Eq. 共14兲兴. As a consequence, the intensities have different finite-size scaling, depending on whether the pair of excited spinons contains the soft spinon or not. We can therefore identify three contributions. Elastic peak. This is the peak with the largest intensity at Q = Me 关shown in blue 共online兲 in Fig. 9共c兲 and cut in intensity to show the other excitations兴. It comes from a pair of 共identical兲 soft spinons 共with wave vector q0 = 0兲 in Eq. 共21兲 and so has energy O共1 / N兲 and intensity proportional to 兩Uq0il0Vq0il0 / 冑N兩2 ⬃ N. By integrating over all frequencies, only this 共zero-frequency兲 peak contributes to the peak of the 共equal-time兲 structure factor, Sxx共Me兲关Fig. 6 共right兲兴. We also note that the peak is absent in the zz response 关Fig. 9共d兲兴, which is expected because the correlations are long ranged in the plane only. Magnon branches. There are three magnon branches 关shown in red 共online兲 in Figs. 9共c兲 and 9共d兲兴, two are “optical” modes, the third one, gapless, is the Goldstone mode of the U共1兲 symmetry. The 共almost兲 flat mode is the weathervane mode which is always gapped in the Schwinger-boson approach, contrary to spin-wave theory,47 and irrespective of S.28 The magnon here consists of a pair of the soft spinon and a spinon of wave vector Q 关see Eq. 共21兲兴 so that the magnon dispersion is the spinon dispersion ␻Q␮ 共Ref. 48兲 and the intensities are of order 兩UqinVq0il0 / 冑N兩2 ⬃ 1. Continuum. As for the spin-liquid phase, there is a continuum of excitations obtained from contributions in Eq. 共21兲 with two spinons both different from the soft mode. Each peak has intensity 兩UqinVkil / 冑N兩2 ⬃ O共1 / N兲 and the sum of them gives a continuum with finite intensity in the thermodynamic limit, which is absent in lowest-order spin-wave theory. All these excitations contribute to the sum rule, N具共Sxi 兲2典 = 兺q兰共d␻ / 2␲兲Sxx共q , ␻兲, given the different density of states. Note that, as expected, the Dzyaloshinskii-Moriya interac-

Dzyaloshinskii, J. Phys. Chem. Solids 4, 241 共1958兲; T. Moriya, Phys. Rev. 120, 91 共1960兲. 2 N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 共1991兲. 3 X. G. Wen, Phys. Rev. B 44, 2664 共1991兲. 4 G. Misguich and C. Lhuillier, in Frustrated Spin Systems, edited by H. T. Diep 共World Scientific, Singapore, 2005兲. 5 O. Cépas, K. Kakurai, L. P. Regnault, T. Ziman, J. P. Boucher, N. Aso, M. Nishi, H. Kageyama, and Y. Ueda, Phys. Rev. Lett. 87, 167205 共2001兲. 1 I.

tion introduces an anisotropy between the in-plane 关Figs. 9共a兲 and 9共c兲兴 and the out-of-plane 关Figs. 9共b兲 and 9共d兲兴 dynamical factors. This anisotropy is visible in the spin-liquid as well as in the Néel-ordered phase. The effect is more spectacular in the latter where the Dzyaloshinskii-Moriya interaction strongly suppresses the low-energy response in the zz channel around the Me point. VI. CONCLUSION

We have obtained the Schwinger-boson mean-field phase diagram for different values of S. The large S limit is in agreement with the semiclassical order by disorder mechanism which selects the 冑3 ⫻ 冑3 state.28 We find that this phase remains stable at small anisotropy in a region which becomes broader and broader when the spin decreases 共and hence quantum-mechanical effects increase兲. It is therefore possible in principle to observe both ordered phases experimentally and first-order transitions between them. However, given the small critical strength, the Q = 0 phase is more likely to occur in a real compound with large enough S and the kagome potassium jarosite 共S = 5 / 2兲 offers such an example.49,50 We have identified a region of the phase diagram 关S ⬃ 0.2, Ansatz 共␲ , 0兲兴 which resembles qualitatively to the exact diagonalization results for the S = 1 / 2 system, where the Dzyaloshinskii-Moriya interaction induces a quantum phase transition between a topological spin liquid and the Q = 0 Néel-ordered phase. This suggests to consider smaller values of the “spin” parameter S as possibly relevant within the Schwinger-boson theory, given that the treatment on average of the constraint leads to enhance 具S2i 典 compared to S共S + 1兲. Within this framework, we could calculate observables such as 共i兲 the cross section of diffuse neutron scattering, with the evolution from broad diffuse scattering to Bragg peaks across the quantum phase transition 共Figs. 6 and 8兲 and 共ii兲 the neutron inelastic cross section which, in addition to Bragg peaks and spin waves, shows a broad continuum in both disordered and ordered phases 共Fig. 9兲, absent in the lowest-order of spin-wave theory. ACKNOWLEDGMENTS

We thank G. Misguich and B. Douçot for numerous enlightening comments on the Schwinger-boson theory.

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bach, Interacting Electrons and Quantum Magnetism 共SpringerVerlag, Berlin, 1994 for the discussion of the 1 / N corrections兲 is to use the Wick theorem:具n2i 典 − 具ni典2 = 具nai典共具nai典 + 1兲 + 具nbi典共具nbi典 + 1兲 + 兩具a2i 典兩2 + 兩具b2i 典兩2 + 2兩具aibi典兩2 + 2兩具a†i bi典兩2. At D = 0, the Hamiltonian is SU共2兲 invariant and we obtain 具n2i 典 − 具ni典2 = 2S共S + 1兲. For D ⫽ 0, 具aibi典 ⫽ 0. 39 At finite N and for sufficiently small boson density, the problem is strictly equivalent to gapped spinons 共the Schwinger bosons兲 interacting with a fluctuating bond field. It has been understood that the low-energy physics of such a system will be that of a lattice gauge theory coupled to charged matter fields. But the nature 共group兲 of the gauge field crucially depends on the lattice geometry and in the present case where the lattice is not bipartite, it should generically be of Z2 type and has two types of phases 共Refs. 2 and 40兲. If the effective Z2 gauge theory is in a deconfined phase 共which should be the case for large enough N兲, the ground state is qualitatively close to the mean-field one, with gapped and unconfined spinons. If the phase is instead confined, the gauge fluctuations are strong and cannot be neglected, the mean-field picture is not valid any more. 40 G. Misguich, arXiv0809 共unpublished兲. Lectures given at the Les Houches summer school on Exact Methods in Low-Dimensional Statistical Physics and Quantum Computing, 2008. 41 For the sake of simplicity we develop only the formulas corresponding to a unique soft mode. This is the case of the Néel order in presence of a Dzyaloshinskii-Moriya interaction: due to the U共1兲 symmetry of the original Hamiltonian, the Goldstone mode is unique. The general case is straightforward but more cumbersome in writing. 42 2 mAF is delicate to normalize because of the fluctuations of the spin length on each site. Fluctuations also increase with the strength of the Dzyaloshinskii-Moriya coupling, making the normalization partly arbitrary. 43 H. Neuberger and T. Ziman, Phys. Rev. B 39, 2608 共1989兲; D. S. Fisher, ibid. 39, 11783 共1989兲; P. Hasenfratz and F. Niedermayer, Zeitschrift für Physik B 92, 91 共1993兲. 44 J. N. Reimers, Phys. Rev. B 46, 193 共1992兲. 45 B. Fåk, F. C. Coomer, A. Harrison, D. Visser, and M. E. Zhitomirsky, EPL 81, 17006 共2008兲. 46 S.-H. Lee, C. Broholm, G. Aeppli, A. P. Ramirez, T. G. Perring, C. J. Carlile, M. Adams, T. J. L. Jones, and B. Hessen, Europhys. Lett. 35, 127 共1996兲. 47 A. B. Harris, C. Kallin, and A. J. Berlinsky, Phys. Rev. B 45, 2899 共1992兲. 48 If q ⫽ 0, the spinon dispersion is folded in the magnetic Bril0 louin zone. 49 A. S. Wills, Phys. Rev. B 63, 064430 共2001兲. 50 For S ⬎ 1 / 2, we note that single-ion anisotropies are also present and may affect the form of magnetic order.

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